Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

learn more… | top users | synonyms

3
votes
1answer
63 views

Compact linear operator

Today in lecture we were told that for a linear compact operator $T$ on an infinite-dimensional Hilbert space with infinite-dimensional range, we have that $\ker(T)^{\perp}$ is infinite-dimensional, ...
1
vote
1answer
31 views

Range of operator always closed. Mistake in argument

Let $A \in L(X,Y)$ be a linear operator between Hilbert spaces and the operator $$\hat{A}: \ker(A)^{\perp} \rightarrow \operatorname{ran}(A)$$ is a restriction of $A$ which is bijective. Now $\ker(A)^{...
2
votes
1answer
22 views

Can a sequence of von Neumann algebras determine a maximal directed set of subalgebras?

Can a von Neumann algebra $A$ have an infinite sequence $A_0 \subset A_1 \subset A_2 \subset ...$ of sub-vN-algebras such that every other sub-vN-algebra $B \subseteq A$ satisfies, for some $n \geq 0$ ...
0
votes
1answer
48 views

How to find the image of an arbitrary element under this operator?

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$ and define the right shift operator to be the linear operator $T \colon H \to H$ such that $T e_n = e_{n+1}$ for $n = 1, 2, ...
0
votes
1answer
31 views

Spectral Measures: Special Spectrum

Problem Given a Hilbert space $\mathcal{H}$. Denote eigenvalues by: $$\sigma_0(N):=\{\lambda\in\mathbb{C}:\mathcal{N}(\lambda-N)\neq(0)\}$$ Then arbitrary sets admit: $$\Lambda\subseteq\mathbb{C}:\...
0
votes
1answer
20 views

Reducing Spaces: Decompostion

This thread is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Regard a decomposition: $$\mathcal{S}_\...
1
vote
1answer
43 views

Spectral Measures: Multi Version (III)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: $$E:\mathcal{B}(...
0
votes
0answers
40 views

Formal decomposition of Hamiltonian into $A A^\ast$

Let $H = -\frac{d^2}{dx^2} + q$. Letting aside consideration of domains, I want to show that $H$ can be formally written as $H = A A^\ast$, where $A = -\frac{d}{dx} + \phi$ with some $\phi$ under the ...
1
vote
1answer
35 views

Spectral Measures: Multi Version (II)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: $$E:\mathcal{B}(...
1
vote
1answer
43 views

Spectral Measures: Multi Version (I)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad N=\int\lambda\mathrm{d}E(\...
2
votes
0answers
20 views

Closed representation of Ladder operators in One Dimensional Second Order Homogeneous Differential Equations

(1) Has anyone published the closed representation of ladder operators for second order differential equations? More specifically the second order differential equation $$ -\partial_x^2\Psi_m(x) + V(...
1
vote
1answer
24 views

Matrix factorization inequality

How does one show that the following matrix factorization inequality holds in $M_{n} (\mathcal{A})^{+}$, $$(a_{i}^{*}a^{*}aa_{j}) \leq ||a^{*}a|| \cdot (a_{i}^{*}a_{j})$$ Notation. Let $M_{n} (\...
1
vote
2answers
351 views

Norm of a left shift operator

Left shift operator is $L:\ell^2\to\ell^2$ defined by $$(x_1,x_2,x_3,x_4,\ldots)\mapsto (x_2,x_3,x_4,\ldots).$$ This is not an isometry apparently, so $\|Lx\|\ne \|x\|$. Does this mean $\|L\|\ne1$?
1
vote
1answer
31 views

Spectral Measures: Adjoint

This thread is only Q&A! (See the hint: SE: Q&A) Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the ...
0
votes
1answer
18 views

Spectral Measures: Normality

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: $$\int|f(\lambda)|^2\...
1
vote
3answers
65 views

Two normal operator that commutes

Suppose $N\in B(H)$ is normal, and $T\in B(H)$ is invertible. Prove that if $TNT^{-1}$ is normal then $N$ commutes with $T^*T$. I can not any idea to prove it, just I know $T^*TNT^{-1}{T^*}^{-1}N^*T^...
1
vote
2answers
54 views

Prove that a functional is convex

Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. We define the functional $\Phi$ as: \begin{equation} \Phi(x)=\frac{1}{2}(Tx,x) \end{equation} My exercise says that $\Phi$ is ...
1
vote
0answers
70 views

Frechet derivative of an operator

Let an operator $T:C[a,b]\to C[a,b]$ be defined as: \begin{equation} (Tu)(x)=\int_{a}^{b}K(x,t)f(t,u(t))dt \end{equation} where $K:[a,b]\times[a,b]\to \mathbb{R}$, $f:[a,b]\times \mathbb{R}\to \...
1
vote
1answer
27 views

Spectral Measures: Boundedness

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: $$\int|f(\lambda)|^2\...
0
votes
1answer
94 views

Spectral Measures: Existence

Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ By the previous threads: $$Z=N\sqrt{(1+N^*N)^{-1}}\quad N=Z\left(\sqrt{1-...
0
votes
1answer
36 views

Spectral Measures: Invertibility

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Regard the domain: $$\int|f(\lambda)|^2\...
0
votes
1answer
53 views

Mourre Adjoint: Approximation

I will provide an answer later... Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And an operator: $$A:\mathcal{H}\to\mathcal{H}:\quad\|A\...
0
votes
1answer
41 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint operator?

Let $H_1$ and $H_2$ be finite-dimensional (real or complex) Hilbert spaces, let $T \colon H_1 \to H_2$ be a linear operator, [Then $T$ can be shown to be bounded] and let $T^* \colon H_2 \to H_1$ ...
1
vote
1answer
74 views

How to prove Cholesky decomposition for positive-semidefinite matrices?

According to Cholesky decomposition $A$ is a Hermitian positive-definite matrix if and only if $A=T^*T$ for some upper triangular matrix $T$. When $A$ is positive-semidefinite we have such ...
0
votes
2answers
65 views

Question about trace class operators

Let $\cal{H}$ be a Hilbert space, $T$ a bounded linear operator on $\cal{H}$, $S$ a trace class operator, then can one verify that $$|Tr(TS)|\leq\|T\|\cdot|Tr(S)|?$$
0
votes
0answers
33 views

Dimension of a subspace of a Hilbert space

Suppose $\{A_n\}$ is a sequence of operators on Hilbert space $H$ such that $\dim (\overline{\operatorname{Im} A_n})\leq \alpha$ where $\alpha\geq N_0$. If $A_n\to A$ uniformly, then $AH=\lim A_n H$....
0
votes
0answers
24 views

Construct a set of anticommutative operators

Suppose that I have a set of infinitely many operators $\mathbb{A} = \{ A_1, \dots , A_n \}$ with $n \mapsto \infty$. All operators satisfy $A_i A_j + A_j A_i = 0$ for all $i,j \in \{ 1, \dots, n \}$....
2
votes
0answers
22 views

Find the inverse of an operator, and determine is it bounded.

I've been doing some similar problems, but I got stuck on this one... and I have a feeling I'm running in circles trying to solve it. Any help appreciated! Problem: We have an operator: $$ A : C[...
1
vote
1answer
35 views

A problem which reverses the definition of a bounded operator

I've encontered a problem that appears simple, almost like it's a definition of a bounded operator, but with a reversed inequality sign... and I can't seem to find my way to a solution. Any ...
1
vote
1answer
45 views

Mourre Adjoint: Regularity

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the domain: $$A\in\mathcal{B}(\mathcal{H}):\quad\...
4
votes
1answer
153 views

Fractional powers of positive self-adjoint operators

Consider two positive unbounded operators $A$ and $B$ densely defined on a Hilbert space $H$ self-adjoint on a domain $\mathcal{D}(A) = \mathcal{D}(B) = H_1$. By the spectral theorem, we can define ...
0
votes
1answer
56 views

Mourre Adjoint: Algebra

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the domain: $$A\in\mathcal{B}(\mathcal{H}):\quad\...
3
votes
0answers
73 views

Pureness of Vector States

How does one show that irreducibility is equivalent to a vector state being pure? In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let $\...
-1
votes
1answer
39 views

${x_n} \to x$ weakly, why does $T{x_n} \to Tx$ weakly? [closed]

If $T \in B(X,Y)$ and ${x_n} \to x$ weakly, why does $T{x_n} \to Tx$ weakly?
2
votes
1answer
73 views

ODE's & PDE's: Homogenous piecing vs Eigenexpansion vs Green functions

I don't know if i'm within rules of the forum to ask this question. If i'm not please comment before downvoting. If you know of a source that answers these questions, please suggest. It would be ...
1
vote
0answers
57 views

Generating fractional taylor series

I was considering the notion of taylor series which posit that the sum $$ \sum_{i=0}^{\infty} \frac{1}{i!} a_ix^i $$ Where: $$ a_i = \frac{d^if}{dx^i}_{x= a} $$ Converge to the function f in a ...
1
vote
0answers
36 views

Riesz representation theorem for $\langle\mathcal A u,v\rangle$.

Let $V$ be a Hilbert space, and let $V^*$ denote its dual space, consisting of all continuous linear functionals from $V$ into the field $\mathbb R$ or $\mathbb C$. If $x$ is an element of $V$, then ...
0
votes
1answer
50 views

Spectral Measures: Constructions

Any constructions welcome!!! Given a Hilbert space $\mathcal{H}$. Regard spectral measures: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ That are additive: $$E\left(\biguplus_kA_k\right)...
4
votes
2answers
66 views

Norm of the operator $T:\ell^2 \to \ell^2$ defined as $(Tx)_1=0, (Tx)_n=-x_n+\alpha x_{n+1}$

Consider the operator $T: \ell^2 \to \ell^2$ defined as $$\begin{cases} (Tx)_1 = 0, \\ (Tx)_n = -x_n + \alpha x_{n+1}, \quad n\ge 2 \end{cases} $$ where $\alpha \in \mathbb{C}$. I want to find ...
0
votes
1answer
40 views

x-momentum operator $p_x$ expressed as multiple of Translation operator

On this page https://en.wikipedia.org/wiki/Rotation_operator_%28quantum_mechanics%29 under "The translation operator," they use Taylor expansion. As part of that proof they state $p_x = ih * dT(0)...
0
votes
1answer
30 views

Convergence of an increasing sequence of operators in a Hilbert space

I am searching for a theorem of the following form: if $T$ is a (unbounded) self-adjoint operator on a Hilbert space $H$ and $(h_n)_n$ a increasing sequence of bounded Borel functions, which converges ...
1
vote
2answers
58 views

Every non-compact Hermitian operator P has an infinite dimensional invariant subspace on which P is bounded from below

I want an explanation of the following statement. If $P$ is a Hermitian operator on Hilbert space and not compact, there exists an infinite-dimensional subspace $M$, invariant under $P$, on which $P$ ...
0
votes
2answers
42 views

Can you have an operator on a vector space such that it is injective but its kernel is not the zero element?

Take any vector space $V$ and an operator $T : V \mapsto V$ Can there exist a $T$ such that it is injective but $\ker T \neq \{0\}$ and equal to some other element instead?
2
votes
2answers
49 views

uniformly convergent subsequence of bounded linear operators on a Hilbert space?

I am working a problem in which we start with a Hilbert space $\mathcal{H}$ and a sequence $\left\{a_n\right\} \subset \mathcal{H}$ with $||a_n|| = 1$. We also assume that $$\lim_{n \to \infty} \left&...
0
votes
1answer
42 views

The real version of the Cuntz algebra

Assume that $H$ is a real separable Hilbert space. Are there two operators $T,S \in B(H)$ which satisfy $$TT^{*}+SS^{*}=1,\;\;T^{*}T=S^{*}S=1$$ where * is the adjoint operator?
0
votes
2answers
100 views

Unboundedness of Laplacian

I am currently considering the following operator ("modified Laplacian"): $T \colon \left( W^{2,2}(\mathbb{R}), \| \cdot \|_{L^2} \right) \longrightarrow \left( L^2(\mathbb{R}), \| \cdot \|_{L^2} \...
0
votes
1answer
32 views

T is invertible if and only if $||(T_{n})^{-1}|| \leq M$.

Let $X$ be a Banach space. Let $(T_{n})$ be a sequence of invertible operators in $B(X)$ and let $T \in B(X)$ be the uniform limit of $(T_{n})$, i.e., $||T_{n}-T||\rightarrow0$ as $n \rightarrow \...
5
votes
0answers
63 views

Infer $Tf=\sum_{n=1}^\infty (f,f_n) f_n$ from frame condition.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that, for $A<B$ and for all $f\in H$ ...
1
vote
0answers
42 views

Eigenvalue-eigenvector equation for an operator

Proof: Given an eigenvalue-eigenvector equation, suppose that the state vector depends on an external parameter, e.g. time, and that over it acts an operator that is the fourth derivative w.r.t. ...
4
votes
1answer
39 views

The (un)boundedness of an involutive operator

This is a question I've been thinking about for a while, for which I do not have a satisfactory answer. Suppose that $T$ is a densely-defined operator on a Hilbert space $\mathcal{H}$ such that $R(T)\...